How Weirs Measure Flow: Rectangular, V-Notch, Cipolletti

Put a plate across an open channel and the water has to climb over it. How high it climbs depends on one thing only: how much water is coming. That makes the upstream water level a flow meter — the oldest one in hydraulics, and still one of the most accurate when it is built and read properly. A thin-plate weir inside the limits of ISO 1438 measures discharge to about 1–2 %, with no moving parts and nothing more than a staff gauge or a level sensor.

The catch is that "how high" depends on the shape of the opening and on how cleanly the water separates from the crest, so there is a different equation for each shape and a short list of installation rules behind every coefficient. Here is what the common methods say about one site — a 1.0 m wide concrete channel with a crest 0.5 m above the bed and 200 mm of head on the gauge — before any of it is explained:

Every number in this article comes from the weir flow calculator, which runs the same functions; the defaults reproduce the first row.

Why the head measures the flow

Take a rectangular opening of width $b$ and look at a thin strip of it at depth $y$ below the upstream water surface. If the water there were simply falling freely from the surface, Bernoulli says it would move at $\sqrt{2gy}$. Integrate that velocity over the strips from the surface down to the crest at depth $h$ and you get the ideal discharge:

• $Q_{ideal}$ — discharge if every streamline fell freely from the surface (m³/s)
• $b$ — width of the opening (m)
• $h$ — head: the height of the undisturbed upstream surface above the crest (m)
• $g$ — gravitational acceleration, 9.80665 m/s²

The 3/2 power is the whole story of weir measurement: discharge grows faster than head, so a modest range of heads covers a wide range of flows, and a small error in head becomes a 1.5 × larger error in flow.

Real water does not fall freely. The streamlines curve as they approach the crest, the sheet of water — the nappe — contracts after it leaves the plate, and there is a velocity in the approach channel that the simple integral ignores. All of that is swept into a discharge coefficient of roughly 0.6, and the differences between the methods below are differences in how carefully that 0.6 is pinned down.

The rectangular sharp-crested weir

Kindsvater-Carter (ISO 1438)

Kindsvater and Carter's 1957 study is the basis of the rectangular-weir method in ISO 1438 and the USBR Water Measurement Manual. Their idea was to keep the coefficient a clean function of geometry by moving the small, awkward effects of viscosity and surface tension into two fixed corrections on the dimensions:

• $C_e$ — effective discharge coefficient, a straight line in $h/P$ whose slope and intercept depend on $b/B$
• $b$ — crest width (m)
• $k_b$ — width correction (m), a few millimetres, tabulated against $b/B$
• $h$ — measured head (m)
• $k_h$ — head correction, 0.001 m
• $P$ — crest height above the approach-channel bed (m)
• $B$ — approach channel width (m)

The two ratios do the work. $b/B$ is the contraction ratio: 1.0 for a suppressed weir whose crest spans the whole channel (no side contraction of the nappe), smaller for a contracted weir with the nappe pulling in from each end. $h/P$ carries the approach velocity: the lower the crest, the faster the water arrives, the more kinetic energy it brings to the crest and the higher the effective coefficient. The ISO table:

The USBR manual prints the same lines in feet-and-seconds units (3.220 and 0.4000 for the suppressed weir, which are 0.602 and 0.075 once the $\tfrac{2}{3}\sqrt{2g}$ is taken out); the calculator interpolates between the tabulated ratios.

For the site above: $h/P = 0.20/0.50 = 0.40$, so $C_e = 0.602 + 0.075 \times 0.40 = 0.632$; the effective width is $1.0 - 0.0009 = 0.9991$ m and the effective head $0.201$ m, giving

The approach velocity at that flow is 0.168 / (1.0 × 0.7) = 0.24 m/s, which is exactly the kind of number the $h/P$ term exists to account for.

The Francis formula

Before Kindsvater and Carter, the standard rectangular-weir equation was the one James Francis fitted to his Lowell canal experiments in the 1850s, and it is still printed in every US manual:

• $1.838$ — the US constant 3.33 ft½/s converted to SI (m½/s); it is $0.623 \times \tfrac{2}{3}\sqrt{2g}$
• $b - 0.2h$ — effective crest width for two end contractions (m); for a suppressed weir use $b$ alone

Francis treats each end contraction as removing 0.1 $h$ of crest, and uses a single fixed coefficient of 0.623. There is no $h/P$ term, so there is no approach-velocity correction, which is why the USBR quotes the formula only for crests at least 2 $h$ above the bed (contracted) or 3 $h$ (suppressed). On the site above it gives 0.1644 m³/s, 2.1 % below the ISO value — and that gap is just the missing approach-velocity term at $h/P = 0.4$. Narrow the crest to 0.6 m in the same 1.0 m channel and the ISO method gives 96.4 L/s ($C_e = 0.600$, $k_b = +3.6$ mm) against Francis's 92.1 L/s, 4.5 % low. Francis is fine as a check and perfectly adequate for a tall-crested contracted weir; for anything to be reported as a measurement, use the ISO form.

The V-notch (triangular) weir

Cut a V into the plate instead of a rectangle and the width of the opening grows with depth, so the integral picks up another power of $h$:

• $\theta$ — full included angle of the notch (degrees); 90° is the usual choice, 20°–100° the calibrated range
• $C_e$ — effective coefficient, 0.578 for a 90° notch, from the Kindsvater-Shen curve (a function of $\theta$ alone when the notch is fully contracted)
• $k_h$ — head correction (m), about 0.9 mm at 90° and larger for narrow notches
• $h$ — head above the vertex of the notch (m)

For 90°, $\tan 45° = 1$ and everything collapses to the familiar $Q \approx 1.38\,h^{5/2}$: at 300 mm of head the ISO form gives 67.8 L/s and the shortcut 68.0.

The 5/2 power is why V-notches own the low-flow end of the range. Compare the two weirs on the site above at small flows. The suppressed rectangular weir's minimum reliable head is 30 mm (below that, surface tension and a nappe that clings to the plate wreck the coefficient), and at 30 mm it passes 9.8 L/s. A 90° notch reads that same 9.8 L/s at a head of 138 mm — 4.6 times more water column to measure, with the same gauge. Run the comparison the other way and one 90° notch covers 1 L/s at 55 mm to 100 L/s at 350 mm; the rectangular crest would need to be read to a fraction of a millimetre at the bottom of that range.

The coefficient curve was measured for fully contracted notches, which in ISO 1438 means $h/P \le 0.4$, $h/B \le 0.2$, $P \ge 0.45$ m, $B \ge 0.9$ m and $0.05 \le h \le 0.38$ m. A 90° notch that is partially contracted (lower, or in a narrow channel) has its own correction in the standard; outside those limits the calculator flags the row rather than silently applying the fully contracted value.

The Cipolletti weir

Cesare Cipolletti's trick, from 1886, was to slope the ends of a contracted rectangular weir at 1 horizontal to 4 vertical. The extra area in the two triangles is just enough to make up for the end contractions, so the weir behaves like a suppressed rectangular one of its bottom width:

• $1.859$ — the USBR constant 3.367 ft½/s in SI (m½/s); approach velocity neglected, accuracy about ±5 %
• $b$ — bottom (crest) width of the trapezoid (m)

A 1.0 m Cipolletti crest at 200 mm gives 166.3 L/s — within 1 % of the suppressed rectangular value, which is the whole point — but it does not fit the 1.0 m channel of the site above: each end needs $2h$ = 0.4 m of clearance for the contractions to form, so that crest wants a channel at least 1.8 m wide. The usual limits are $h \ge 0.06$ m, $h \le b/3$, a crest at least $2h$ above the bed and side contractions of at least $2h$.

The broad-crested weir

Make the crest long in the flow direction — a concrete sill rather than a plate — and the water no longer springs clear of it. Over a crest at least twice the head in length (Bos: $H/L \le 0.5$), the streamlines become parallel and the flow passes through critical depth, where the specific energy is a minimum. At critical flow the depth is two thirds of the total head and the velocity is $\sqrt{g\,y_c}$, so the discharge is fixed by the head with no empirical coefficient at all in the ideal case:

• $y_c$ — critical depth over the crest, $\tfrac{2}{3}H$ (m)
• $H$ — total head: measured head plus the approach velocity head $V^2/2g$ (m)
• $1.705$ — $(2/3)^{3/2}\sqrt{g}$, the origin of the "C = 1.7" in textbook formulas

The real coefficient is a little below 1.0 because a boundary layer grows along the crest and the side walls and displaces the flow. The 1976 edition of Bos's Discharge Measurement Structures, quoting BS 3680-4B (the form behind ISO 4374), gives it for a round-nosed horizontal crest as

• $x$ — boundary-layer factor: 0.005 for a well-finished concrete field structure, 0.003 for a laboratory weir
• $L$ — crest length in the flow direction (m), between 2 and 20 $H$; the crest must also be at least 0.30 m, $H$ and $L/5$ wide
• $r$ — radius of the rounded upstream edge (m), at least 0.11 $H$

On the site above, a 1.0 m wide sill 0.6 m long with a 0.1 m nose radius has $C_d = 0.976$ and, once the approach velocity head is added to the 200 mm (total head 0.2024 m), passes 151.5 L/s — about 10 % less than the thin plate, because the thin plate's contracting nappe actually carries more flow per unit head than a crest that forces the water through critical depth. Bos quotes the uncertainty of the coefficient as ±2(21 − 20 $C_d$) %, which is ±3 % here. (The 1989 edition replaces this coefficient by the simpler $C_d = 0.93 + 0.10\,H/L$ with an uncertainty of ±(3|$H/L$ − 0.55| + 4) %; on this sill it gives 0.964 against 0.976, but at low $H/L$ on long crests the two forms differ by up to about 10 %, so the calculator names the edition it uses.) A square-edged sill separates at the upstream corner and runs lower, around 0.85 at low $H/L$; it needs a measured coefficient.

The rules that decide the accuracy

Each coefficient above was measured under conditions, and the conditions are not optional. The ones that go wrong in the field:

1. Where the head is read. The surface starts to draw down well before the crest (the figure shows it). Read the level at least 4 × the maximum head upstream — 2–3 × for a broad-crested weir — in a stilling well, never on the plate. A gauge on the plate reads the dropped surface and under-reports.
2. The gauge zero. It must be set to the crest level or the notch vertex. A 2 mm zero error is 1.5 % of flow at 200 mm of head on a rectangular weir and 2.5 % on a V-notch, which is already more than the coefficient's own uncertainty.
3. A sharp crest. A thin-plate weir's crest is 1–2 mm thick, square on the upstream side and bevelled downstream so the nappe springs free. Rust, a rounded edge or a thick plate turns it into a short broad-crested weir with a different, unknown coefficient.
4. An aerated nappe. The sheet of water must fall with air beneath it. On a suppressed weir the side walls trap that air, the pocket is carried away, the nappe clings to the plate and the flow rises several percent above the equation. Vent it.
5. Free discharge. The tailwater must stay below the crest — ISO 1438 asks for at least 50 mm of clearance. Once the tailwater rises over the crest the downstream water backs the flow up: at a given upstream head a submerged weir passes less than the free-flow equation says, so the equation over-reads. Villemonte's correction for an existing installation is $Q_s = Q_1 \left[1 - (h_2/h_1)^n\right]^{0.385}$, where $Q_1$ is the free-flow discharge at the upstream head $h_1$, $h_2$ is the tailwater head above the crest and $n$ is the exponent of the free-flow equation (1.5 rectangular, 2.5 V-notch); it is approximate, and a weir that submerges at high flow has been built in the wrong place.
6. A quiet approach. A straight, uniform channel for at least 10 widths upstream, no bends, drops or inlets close to the weir, and velocities low enough for a symmetrical profile. The $h/P$ term handles the magnitude of the approach velocity, not a skewed distribution.
7. Stay inside the limits. Minimum heads (30 mm rectangular, 50 mm V-notch, 60 mm Cipolletti), minimum crest heights, minimum side contractions, maximum $h/P$. The calculator computes outside them but flags every row that is outside, because the number is then an extrapolation.

Choosing the weir

Whatever the shape, the coefficient is the smaller part of the uncertainty budget. The head gauge, its zero, the aeration and the approach channel are the measurement; the formula only turns a level into a number. The numbers make the point: flow scales with $h^{3/2}$, so a head error reaches the flow at 1.5 times its relative size. A 2 mm zero or reading error at 200 mm of head is 1 % of head and 1.5 % of flow; combined with a 1 % coefficient uncertainty that is about 1.8 %, the figure ISO quotes. The same 2 mm at 50 mm of head is 4 % of head and 6 % of flow, and the coefficient no longer matters. A V-notch raises the exponent to 2.5 but also the head at a given flow, which is why it keeps the small flows measurable. The weir flow calculator applies the ISO 1438, USBR and Bos forms above with their corrections, solves the inverse problem (the head a given flow will produce, for sizing a new weir), prints the rating table and flags every limit — and the engineering unit converter handles the L/s, m³/h and cfs the different manuals insist on. Where the measured flow enters a pipe, SimuPipe takes over for the pressure side.